Codimension 2
From Mathematics
(→Chow groups and Chern classes) |
(→Cech cohomology and the Hodge conjecture in codim 1) |
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| Line 90: | Line 90: | ||
* Review of Cech cohomology and the long exact sequence, explicit construction for the <math>\delta</math> homomorphism. Treat the analytic and Zariski topology parallel. | * Review of Cech cohomology and the long exact sequence, explicit construction for the <math>\delta</math> homomorphism. Treat the analytic and Zariski topology parallel. | ||
* Classification of vector bundles by Cech-cocycles, Theorem I.5.10<ref name="Wei13"/>. | * Classification of vector bundles by Cech-cocycles, Theorem I.5.10<ref name="Wei13"/>. | ||
| − | * Mention also that for a ''projective'' complex algebraic variety <math>X</math>, complex analytic vector bundles, classified by <math>H^1(X^{an}, \mathcal{O}_X^{an})</math> (analytic Cech cohomology), are the same as algebraic vector bundles on <math>X</math>, classified by <math>H^1(X, \mathcal{O}_X)</math> (Zariski Cech cohomology). | + | * Mention also that for a ''projective'' complex algebraic variety <math>X</math>, complex analytic (rank n) vector bundles, classified by <math>H^1(X^{an}, \mathrm{GL}_n(\mathcal{O}_X^{an}))</math> (analytic Cech cohomology), are the same as algebraic (rank n) vector bundles on <math>X</math>, classified by <math>H^1(X, \mathrm{GL}_n(\mathcal{O}_X))</math> (Zariski Cech cohomology). |
* The exponential sequence | * The exponential sequence | ||
<math> 0 \rightarrow 2\pi i \mathbb{Z}_X \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^* \rightarrow 1</math> | <math> 0 \rightarrow 2\pi i \mathbb{Z}_X \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^* \rightarrow 1</math> | ||
